The Picard iteration method is used to study the existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays. Several sufficient conditions are specified to ensure that the equation has a unique solution. First, the stochastic Volterra-Levin equation is transformed into an integral equation. Then, to obtain the solution of the integral equation, the successive approximation sequences are constructed, and the existence and uniqueness of solutions for the stochastic Volterra-Levin equation are derived by the convergence of the sequences. Finally, two examples are given to demonstrate the validity of the theoretical results.
As stochastic modeling is used in the fields such as physics, economics, chemistry, and scholars have paid more and more attention to stochastic differential equations. Therefore, the existence and uniqueness of solutions of the equation have become a hot topic in recent years. The Volterra equation is a significant differential equation, which has been applied to the circulating fuel nuclear reactor, the neural networks, the population projection and others. In 1928, Volterra  first proposed the Volterra equation, i.e.,
and Levin  obtained the asymptotical stability of (1). Burton investigated the stability of equation (1) by the contraction mapping principle in . Zhao and Yuan  considered 3/2-stability of a generalized Volterra-Levin equation. The discrete Volterra equation describing the evolutionary process of the population was recently investigated in .
To analyze the Volterra equation, Levin  used the limited condition that is pretty hard to be checked in practical application, , and the author also required that the function has good properties, such as , and for any . Although the conditions of were simplified by averages in , there were still more requirements for the function . In this paper, the constrains of and will be weaken in the stochastic Volterra-Levin equation with variable delays.
Let be a complete probability space equipped with some filtration satisfying the usual conditions, that the filtration is right continuous and contains all P-null sets. Let denote a standard Brownian motion defined on . We investigate the existence and uniqueness of solutions for the stochastic Volterra-Levin equations with variable delays, i.e.,
where and are known functions satisfying certain conditions, the constant , , and , and are the variable delays, satisfying .
Scholars have become increasingly interested in the stochastic Volterra-Levin equations. The equation has been applied to many special research fields, such as the population model of spatial heterogeneity , the predator-prey model , and the nonautonomous competitive model . After reviewing and sorting out the literature, it is found that most scholars currently use the principle of contraction mapping to explore the equation. For example, Luo  analyzed the exponential stability of the classical stochastic Volterra-Levin equations. Zhao et al.  investigated the mean square asymptotic stability of the generalized stochastic Volterra-Levin equations, which improved the results in . Li and Xu  demonstrated the existence and global attractiveness of periodic solutions for impulsive stochastic Volterra-Levin equations. In this paper, the Picard iteration method is directly used to prove the existence and uniqueness of solutions of the stochastic Volterra-Levin equations with variable delays, which can give a more intuitive approximate solution. Recently, for the case without delay, Jaber  proved the weak existence and uniqueness of affine stochastic Volterra equations. Dung  revealed Itô differential representation of the stochastic Volterra integral equations. For the case of constant delay, Guo and Zhu  used this approximate method to prove the existence of solutions of stochastic Volterra-Levin equations. Some delay Volterra integral problems on a half-line were analyzed in . The qualitative properties of solutions of nonlinear Volterra equations without random disturbance were investigated in . However, there are only a few results of the stochastic Volterra equations with variable delay.
Generally, a time delay is inevitable and variable in practical application, and the future state of an existing system depends not only on the current state of the systems but also on the past [17,18,19]. When the function in equation (2), Benhadri and Zeghdoudi  applied the variable delays to the Volterra-Levin equation with Poisson jump and obtained the mean square stability by the fixed-point theory. The authors in  discussed the linear discrete Volterra equation with infinite delay when the function , which means there are not any random noises. In this paper, we will investigate the Volterra-Levin equation with the variable delays and the standard Brownian motion in more general conditions. Moreover, the Picard successive approximation method is used to prove the existence and uniqueness of the solution in some sufficient conditions. Compared with  and , these conditions are easier to be verified.
The rest of this paper is organized as follows. In Section 2, some necessary conditions and lemmas are established. In Section 3, the existence and uniqueness of solutions are proved. In Section 4, two examples are given to demonstrate the validity of the main results.
2 Assumptions and lemmas
To obtain the existence and uniqueness of the solutions for equation (2), the following assumptions are given in this paper.
, and there exists a constant , such that .
, and .
There is a positive constant , such that for all .
Assumptions – are some common conditions for studying the Volterra-Levin equations. For instance, Luo  discussed the exponential stability for classical stochastic Volterra-Levin equations on Assumptions . Zhao et al.  studied the mean square asymptotic stability of a class of generalized nonlinear stochastic Volterra-Levin equations on similar assumptions. Assumption is the Lipschitz condition, which is the core condition for ensuring the existence and uniqueness of solutions for the initial value problem.
Now, we transform (2) into the following form by using the properties of integrals.
Assuming that – are established, equation (2) can be transformed into
Let , it is obtained that from Assumptions and . Using
Equation (2) can be transformed into
The two sides of the aforementioned equation are multiplied by , and then integral from 0 to , using the distribution integral method and the following formula:
Two sides of the aforementioned equality are multiplied by , and using
We can obtain equation (3).□
The method of transforming the stochastic differential equation into the integral equation, has been widely used. When the function is independent of the variable , Luo studied the exponential stability for a class of stochastic Volterra-Levin equations by using the method in . Zhao et al. investigated the mean square asymptotic of the generalized nonlinear stochastic Volterra-Levin equations . Based on the semigroup of operators, Yang et al. transformed the heat conduction equation into the fractional Volterra integral equation in . In this lemma, due to the appearance of variable delays, we need to deal with it more precisely.
3 Existence and uniqueness
Picard iteration is the most commonly used method in the proof of the existence of solutions to the stochastic equations [20,21, 22,23]. In this paper, the existence and uniqueness of solutions for equation (2) are proved by the Picard iteration method. An important characteristic of this method is that it is constructive, and the bounds on the difference between iterates and the solutions are easily available. Such bounds are not only useful for the approximation of solutions but also necessary in the study of qualitative properties of solutions.
Now, let’s briefly summarize this idea of the Picard iteration method. To obtain the solution for a class of integral equation , Picard successive approximation sequences are constructed as follows.
If the sequences converge uniformly to a continuous function in some interval , then we may past to the limit in both sides of the aforementioned equation to obtain
So that is the desired solution.
Suppose that assumptions – hold, then equation (2) has a unique solution in .
The Picard iteration method is used in the proof of this theorem, and using Lemma 2.1, we construct the Picard iteration sequences.
(1) We first verify the mean square boundness of , so we only need to prove is bounded.
It is obvious that for . Suppose is bounded, we begin to prove is bounded. Using the formula (9), it obtains .
Using Assumptions and , it obtains
Further, by using Hölder inequality, we obtain
From Assumptions and , we know
(2) Verifying the mean square continuity of
Suppose and is sufficiently small, we obtain the properties as follows.
By Itô integration, we have
So , the mean square continuity of is verified.
(3) This part proves the convergence of sequences .
By using the similar method of Step (1), we have
By Chebyshev inequality, we obtain
From Assumption , we have . By Borel-Cantelli lemma, it follows that there exists a positive integer for almost all , satisfying
for any .
Next we show that are uniformly convergent on . Since can be regarded as the partial sum of function series , as well as , it follows that are uniformly convergent on by using the convergence of constant series and Weierstrass’ discriminance.
Let be the sum function, it obtains the function sequences converge uniformly to on . Considering are continuous and compatible, we obtain the sum fucntion that is also continuous and compatible. Using inequality (15), are the Cauchy sequences in , so
On the other hand, by inequality (15), we have
So by Assumption , it obtains .
(4) This part proves that is a solution for equation (2)
After simple calculation, we have
Using Hölder inequality, it follows